3.1 Exercises

20 points · 4 extra credit

Introduction to Determinants

1. 1. (2 pts) Compute the determinant of $\begin{bmatrix} 5 & 3 \\ -2 & 4 \end{bmatrix}$.

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2. 2. (2 pts) Compute $\det A$ by a cofactor expansion across the first row, where

   $A = \begin{bmatrix} 1 & 0 & 2 \\ 3 & -1 & 4 \\ 2 & 1 & 5 \end{bmatrix}.$

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3. 3. (3 pts) Compute the determinant of

   $B = \begin{bmatrix} 2 & 0 & 0 & 4 \\ 1 & 3 & 0 & 0 \\ 0 & 5 & -1 & 0 \\ 6 & 0 & 0 & 2 \end{bmatrix}$

   by choosing a row or column that keeps the arithmetic short. State which one you expanded along and why.

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4. 4. (2 pts) Without doing any expansion, write down $\det T$ for the triangular matrix

   $T = \begin{bmatrix} 4 & 9 & -3 \\ 0 & 2 & 7 \\ 0 & 0 & 5 \end{bmatrix}.$

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5. 5. (3 pts) For the matrix $A$ of Problem 2, recompute $\det A$ by expanding along the second column. Confirm you get the same value, and explain in one sentence why any row or column must give the same answer.

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6. 6. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) The $(i,j)$-cofactor of a matrix equals the determinant of its $(i,j)$ submatrix $A_{ij}$.

   (b) If a square matrix has a row of zeros, its determinant is $0$.

   (c) $\det A$ is defined only for square matrices.

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7. 7. (3 pts) Let $A$ be a $3\times 3$ matrix with a row of all zeros. Show, using a cofactor expansion, that $\det A = 0$.

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8. 8. (2 pts) Find all values of $t$ for which $\det\begin{bmatrix} 2-t & 1 \\ 1 & 2-t \end{bmatrix} = 0$.

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9. 9. (4 pts ••) Prove that the determinant of an $n\times n$ upper-triangular matrix equals the product of its diagonal entries. Use induction on $n$ together with a cofactor expansion along the first column.

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